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The category of abelian varieties over Fq is shown to be anti-equivalent to a category of Z-lattices that are modules for a non-commutative pro-ring of endomorphisms of a suitably chosen direct system of abelian varieties over Fq. On full subcategories cut out by a finite set w of conjugacy classes of Weil q-numbers, the anti-equivalence is represented by what we call w-locally projective abelian varieties.
We consider ground state solutions u ∈ H2(RN) of biharmonic (fourth-order) nonlinear Schrodinger equations of the form ¨2u + 2au + bu − |u| p−2u = 0 in RN with positive constants a, b > 0 and exponents 2 < p < 2∗, where 2∗ = 2N N−4 if N > 4 and 2∗ = ∞ if N ≤ 4. By exploiting a connection to the adjoint Stein–Tomas inequality on the unit sphere and by using trial functions due to Knapp, we prove a general symmetry breaking result by showing that all ground states u ∈ H2(RN) in dimension N ≥ 2 fail to be radially symmetric for all exponents 2 < p < 2N+2 N−1 in a suitable regime of a, b > 0. As applications of our main result, we also prove symmetry breaking for a minimization problem with constrained L2-mass and for a related problem on the unit ball in RN subject to Dirichlet boundary conditions.
Geometry is part of the core of mathematics. It has been relevant ever since people have interacted with nature and its phenomena. Geometry’s relevance to the teaching and learning of mathematics can be emphasized, too. Nevertheless, a current potential shift in the topics of mathematics education to the detriment of geometry might be emerging. That is, other topics related to mathematics are seeming to grow in importance in comparison to geometry. Despite this, or perhaps because of it, geometry is an important component of current research in mathematics education. In the literature review, we elaborate relevant foci on the basis of current conference proceedings. By means of about 50 journal articles, five main topics are elaborated in more detail: geometric thinking and practices, geometric contents and topics, teacher education in geometry, argumentation and proof in geometry, as well as the use of digital tools for the teaching and learning of geometry. Conclusions and limitations for current and future research on geometry are formulated at the end of the article. In particular, the transfer to the practices of geometric teaching is explored on the basis of the elaborated research findings in order to combine both aspects of the teaching and learning of geometry.
The free energy of TAP-solutions for the SK-model of mean field spin glasses can be expressed as a nonlinear functional of local terms: we exploit this feature in order to contrive abstract REM-like models which we then solve by a classical large deviations treatment. This allows to identify the origin of the physically unsettling quadratic (in the inverse of temperature) correction to the Parisi free energy for the SK-model, and formalizes the true cavity dynamics which acts on TAP-space, i.e. on the space of TAP-solutions. From a non-spin glass point of view, this work is the first in a series of refinements which addresses the stability of hierarchical structures in models of evolving populations.
Motivated by the question of the impact of selective advantage in populations with skewed reproduction mechanisms, we study a Moran model with selection. We assume that there are two types of individuals, where the reproductive success of one type is larger than the other. The higher reproductive success may stem from either more frequent reproduction, or from larger numbers of offspring, and is encoded in a measure Λ for each of the two types. Λ-reproduction here means that a whole fraction of the population is replaced at a reproductive event. Our approach consists of constructing a Λ-asymmetric Moran model in which individuals of the two populations compete, rather than considering a Moran model for each population. Provided the measure are ordered stochastically, we can couple them. This allows us to construct the central object of this paper, the Λ−asymmetric ancestral selection graph, leading to a pathwise duality of the forward in time Λ-asymmetric Moran model with its ancestral process. We apply the ancestral selection graph in order to obtain scaling limits of the forward and backward processes, and note that the frequency process converges to the solution of an SDE with discontinuous paths. Finally, we derive a Griffiths representation for the generator of the SDE and use it to find a semi-explicit formula for the probability of fixation of the less beneficial of the two types.
Highlights
• We study dormancy in the ‘rare mutation’ regime of stochastic adaptive dynamics.
• We first derive the polymorphic evolution sequence, based on prior work.
• Our evolutionary branching criterion extends a result by Champagnat and Méléard.
• In a classical model dormancy can favour evolutionary branching.
• Dormancy also affects several more population characteristics.
Abstract
In this paper, we investigate the consequences of dormancy in the ‘rare mutation’ and ‘large population’ regime of stochastic adaptive dynamics. Starting from an individual-based micro-model, we first derive the Polymorphic Evolution Sequence of the population, based on a previous work by Baar and Bovier (2018). After passing to a second ‘small mutations’ limit, we arrive at the Canonical Equation of Adaptive Dynamics, and state a corresponding criterion for evolutionary branching, extending a previous result of Champagnat and Méléard (2011).
The criterion allows a quantitative and qualitative analysis of the effects of dormancy in the well-known model of Dieckmann and Doebeli (1999) for sympatric speciation. In fact, quite an intuitive picture emerges: Dormancy enlarges the parameter range for evolutionary branching, increases the carrying capacity and niche width of the post-branching sub-populations, and, depending on the model parameters, can either increase or decrease the ‘speed of adaptation’ of populations. Finally, dormancy increases diversity by increasing the genetic distance between subpopulations.
For genus g=2i≥4 and the length g−1 partition μ=(4,2,…,2,−2,…,−2) of 0, we compute the first coefficients of the class of D¯¯¯¯(μ) in PicQ(R¯¯¯¯g), where D(μ) is the divisor consisting of pairs [C,η]∈Rg with η≅OC(2x1+x2+⋯+xi−1−xi−⋯−x2i−1) for some points x1,…,x2i−1 on C. We further provide several enumerative results that will be used for this computation.
For genus g=2i≥4 and the length g−1 partition μ=(4,2,…,2,−2,…,−2) of 0, we compute the first coefficients of the class of D¯¯¯¯(μ) in PicQ(R¯¯¯¯g), where D(μ) is the divisor consisting of pairs [C,η]∈Rg with η≅OC(2x1+x2+⋯+xi−1−xi−⋯−x2i−1) for some points x1,…,x2i−1 on C. We further provide several enumerative results that will be used for this computation.
Affine Bruhat--Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of PGL parametrizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite-dimensional vector space, up to homothety. It has first been studied by Goldman and Iwahori as a piecewise-linear analogue of symmetric spaces. The space of seminorms compactifies the space of norms and admits a natural surjective restriction map from the Berkovich analytification of projective space that factors the natural tropicalization map. Inspired by Payne's result that the analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized linear embeddings ι:Pr↪Pn and prove a faithful tropicalization result for compactified linear spaces. The space of seminorms is in fact the tropical linear space associated to the universal realizable valuated matroid.
We use recent results by Bainbridge–Chen–Gendron–Grushevsky–Möller on compactifications of strata of abelian differentials to give a comprehensive solution to the realizability problem for effective tropical canonical divisors in equicharacteristic zero. Given a pair (Γ,D) consisting of a stable tropical curve Γ and a divisor D in the canonical linear system on Γ, we give a purely combinatorial condition to decide whether there is a smooth curve X over a non-Archimedean field whose stable reduction has Γ as its dual tropical curve together with an effective canonical divisor KX that specializes to D.